SGU 185. Two shortest

time limit per test: 0.50 sec.

memory limit per test: 4096 KB

input: standard input

output: standard output

Yesterday Vasya and Petya quarreled badly, and now they don't want to see each other on their

way to school. The problem is that they live in one and the same house, leave the house at the

same time and go at the same speed by the shortest road. Neither of them wants to change their

principles, that is why they want to find two separate shortest routes, which won't make them go

along one road, but still they can meet at any junction. They ask you to help them. They number

all the junctions with numbers from 1 to N (home and school are also considered as junctions).

So their house has the number 1 and the school has the number N, each road connects two

junctions exactly, and there cannot be several roads between any two junctions.

Input The first line contains two integer numbers N and M (2<=N<=400), where M is the number of

roads Petya and Vasya noticed. Each of the following M lines contains 3 integers: X, Y and L

(1<=X, Y<=N, 1<=L<=10000), where X and Y - numbers of junctions, connected by the road

and L is the length of the road.

Output Write to the first line numbers of the junctions in the way they passed them on the first route.

Write to the second line numbers of the junctions in the way they passed them on the second

route. If it is impossible to help guys, then output "No solution".

Sample test(s)

Input 6 8

1 2 1

3 2 1

3 4 1

1 3 2

4 2 2

4 5 1

5 6 1

4 6 2

Output 1 3 4 5 6

1 2 4 6

SGU 226. Colored graph

time limit per test: 0.50 sec.

memory limit per test: 4096 KB

input: standard

output: standard

You are given an oriented graph. Each edge of the graph is colored in one of the three colors.

Your task is to find the length of the shortest path from the first vertex to the N-th. Note that any

two successive edges in the path can't have the same color.

Input The first line of the input file consists of two integers N and M (2 <= N <= 200; 0 <= M <=

N*N). Next M lines contain descriptions of the edges. Each edge description is a list of three

integers X, Y, C (1 <= X, Y <= N, 1 <= C <= 3), where X is the starting vertex of the edge, Y is

the finishing vertex and C is the color of the edge.

Output Output the length of the shortest path between the first and the N-th vertexes. Output "-1" if the

path doesn't exist.

Sample test(s)

Input

Test #1

4 4

1 2 1

2 3 2

3 4 3

2 4 1

Test #2

3 2

1 2 1

2 3 1

Output

Test #1

3

Test #2

-1

SGU 298. King Berl VI

Time limit per test: 4 second(s)

Memory limit: 65536 kilobytes

input: standard

output: standard

King Berl VI has N daughters and no sons. During his long life he gave a number of promises to

his daughters. All the promises have the following wording: "daughter Xi, I promise to give you

dowry not less than Ci burles more than to daughter Yi", where i represents the number of the

promise. Before his death the king decided to give some amount of money to each daughter. As

far as he was the fair king, he decided to fullfill all his promises. But he was not only fair but

also very greedy, he decided that he can give negative amount of burles as a dowry (i.e. daughter

should pay this amount of burles to the treasury). Because of his born greed and by advice of the

minister of finances, he made a decision that absolute value of each dowry should not exceed

10000 burles and the difference between dowry of the oldest and of the youngest daughters

should be as small as possible (note, this value can be negative).

I.e. if the dowry given to the i-th daughter is Ai, folllowing conditions should be satisfied:

-10000≤ Ai≤ 10000

AN - A1 should be minimal

Input The fist line of the input file contains two integers numbers N and M (2≤ N≤ 10000; 0≤ M≤

100000), where N is the number of daughters and M is the number of promises. The following M

lines contain the description of promises in the following form: Xi, Yi, Ci

( ). The youngest daughter has the number one, the

oldest — N. Each pair Xi, Yi can appear in the input several times.

Output Write to the output number -1 if there is no solution for the problem (i.e. there is no sequence of

N integers which satisfies all described above requirements). Write to the output N integer

numbers — the amount of dowry of each daughter in burles, if solution exists. If there are

several solutions output any of them.

Example(s)

sample input sample output

4 5

2 1 1

3 1 2

3 2 3

4 2 1

4 3 2

-3 -2 1 3

sample input sample output

2 2

1 2 0

2 1 0

-7 -7

XI Olimpiad in Informatics 2003/2004

Task: WSC

East-West

III stage competition

Source file: wsc.*

Memory limit: 32 MB

International agreements signed by the Awfully Vast State impose on it transit obligations - it

has to enable the transport of nuclear waste from its eastern neighbours power plants to recycling

facilities in the West, by means of its railway system. Ecological reasons impose such traffic

organization, that the waste-carrying trains leave the territory of the state as quickly as possible.

The railway network in this country has a very peculiar structure. It consists of n cities - railway

junctions and n-1 two-way railway track segments, connecting the junctions. Transport is

possible between each pair of cities. Furthermore, there is a section of the railway track, whose

ends are not border cities and every connection from the eastern to the western border has to lead

through that very section.

All waste-carrying trains arrive at the eastern border on the same day, before dawn, at distinct

checkpoints, however. For safety reasons the trains only move during the day. Only a single

waste-carrying train can move on a given track section at a time, but an unlimited amount of

them can wait at a junction. It takes one day for a train to traverse a section of the track. The

traffic has to be organized in such a way, that allows each waste-carrying train to reach a distinct

destination on the western border.

How many days, at least, do the waste-carrying trains have to spend on the territory of the

Awfully Vast State?

Task

Your task is to write a programme which:

reads from the standard input a description of the railway network and border

checkpoints at which the waste-carrying trains have arrived,

finds the minimal number of days the transit has to last,

writes the solution to the standard output.

Input

The first line of the input contains three integers 1 < n, w, z < 106, n >= w+z+2, separated by

single spaces. n denotes the number of junctions (which are labelled with 1,...,n), while w and z

denote the number of border checkpoints on the eastern and western border, respectively. The

checkpoints on the eastern border are labelled with 1,...,w, while those on the western border

with n-z+1,...,n.

In the following n-1 lines there is a decription of the railway network. Each line contains two

distinct integers, 1 < a,b < n, separated by a single space. They denote junctions connected by a

section of the railway.

The n+1st line contains a single integer p, 1 < p < w, 1 < p < z, denoting the number of waste-

carrying trains. In the next (and last) line of the input there are p distinct integers, all of which

are not greater than w, separated by single spaces. These are the numbers of checkpoints on the

eastern border, at which the waste-carrying trains have arrived.

Output

The first and only line of the output should contain exactly one integer, denoting the minimal

amount of days the waste have to spend on the territory of the state.

Example

For the input data: 9 2 3

1 3

2 3

4 3

4 5

4 6

7 4

5 8

9 6

2

1 2

the correct outcome is: 4

The arrows denote the movement of the trains in consecutive days for one of the optimal

organizations of the railway traffic - a loop means that on a given day the train waited at the

junction.

BalticOI 2008 Mafia

Memory limit: 32 MB

The police in Byteland got an anonymous tip that the local mafia bosses are planning a big

transport from the harbour to one of the secret warehouses in the countryside. The police knows

the date of the transport and they know that the transport will use the national highway network.

The highway network consists of two-way highway segments, each segment directly connecting

two distinct toll stations. A toll station may be connected with many other stations. A vehicle can

enter or exit the highway network at toll stations only. The mafia transport is known to enter the

highways at the toll station closest to the harbour and leave it at the toll station closest to the

warehouse (it will not leave and re-enter the highways in between). Special police squads are to

be located in selected toll stations. When the transport enters a toll station under surveillance, it

will be caught by the police.

From this point of view, the easiest choice would be to place the police squad either at the entry

point or the exit point of the transport. However, controlling each toll station has a certain cost,

which may vary from station to station. The police wants to keep the overall cost as low as

possible, so they need to identify a minimal controlling set of toll stations, which satisfies the

two conditions:

all traffic from the harbour to the warehouse must pass through at least one station from

that set,

the cost of monitoring these stations (i.e. the sum of their individual monitoring costs) is

minimal.

You may assume that it is possible to get from the harbour to the warehouse using the highways.

Task Write a program, that:

reads the description of the highway network, the monitoring costs and the locations of

the entry and exit points of the transport from the standard input,

finds a minimal controlling set of toll stations,

writes this set to the standard output.

Input

The first line of the standard input contains two integers and ( ,

) - the number of toll stations and the number of direct highway segments. The

toll stations are numbered from to .

The second line contains two integers and ( , ) - the numbers of the toll

stations closest to the harbour and to the warehouse, respectively.

The following lines describe the monitoring costs. The -th of these lines (for )

contains one integer - the monitoring cost of the -th station (which is positive number not

exceeding ).

The following lines describe the highway network. The -th of these lines (for )

contains two integers and ( ), indicating that there is a direct highway segment

between toll stations numbered and . Each highway segment is listed once.

Additionally, in test cases worth about 40 points, .

Output

The only line of the output should contain the numbers of toll stations in a minimal controlling

set, given in increasing order, separated by single spaces. If there is more than one minimal

controlling set, your program may output anyone of them.

Example For the input data: 5 6

5 3

2

4

8

3

10

1 5

1 2

2 4

4 5

2 3

3 4

the correct result is: 1 4

The figure shows the highway network with the toll station numbers (in the upper-left corners)

and the monitoring costs. Stations number 1 and 4 constitute the minimal controlling set with

total controlling cost 5.

BalticOI 2008 Day 1 Gates Memory limit: 96 MB

After many years of working as a software developer you have decided to try something entirely

different, and started looking at random job offers. The one that really caught your eye was a job

in fish farming (a form of aquaculture). 'Cool!', you thought, and besides, fish are nice creatures.

So you applied, got accepted, and today is your first day at work.

Your boss has already assigned you a task. You have to isolate one water reservoir from another.

After looking at some schemes you've been given, here's what you've figured out.

The two water reservoirs are connected by several channels. Each channel has two gates. The

channel is open when both gates are open, and is closed otherwise. The gates are controlled using

switches. The same switch may operate several gates, but each gate is operated by exactly one

switch. It is possible that both gates on a channel are controlled by the same switch and that a

switch controls no gates.

Example with three channels and two switches.

The switch may operate the gate in one of two ways:

the gate is open when the switch is on, and is closed when the switch is off,

the gate is closed when the switch is on, and is open when the switch is off.

After playing a bit with the switches you suddenly realize that your programming experience will

come in very handy. Write a program that, given the configuration of gates and switches,

determines whether it is possible to close all channels, and if it is, then finds a state of every

switch in one such valid configuration.

Input

The first line of the standard input contains two integers and

, the number of channels and switches respectively. Switches are numbered

from to . Additionally, in test cases worth at least points, will not exceed and will

not exceed .

The following lines describe channels, each channel is described by a separate line containing

four integers: , , , . Numbers and represent switches ( ) that operate gates

of this channel. Numbers and can be either or and correspond to the described operation

modes: means that the gate is closed if and only if the switch is off and means that

the gate is closed if and only if the switch is on.

Output

If it is possible to close all the channels, the standard output should contain lines. Line should

contain , if switch should be off, and if switch should be on. If there are many possible

solutions, your program may output any of them.

If it is impossible to close all channels, your program should output one line, containing a single

word IMPOSSIBLE.

Example

For the input data: 3 2

1 0 2 1

1 0 2 0

1 1 2 1

the correct result is: 0

1

and for the input data: 2 1

1 0 1 0

1 1 1 1

the correct result is: IMPOSSIBLE

The first example corresponds to the picture from the task description.

SGU 381. Bidirected Graph

Time limit per test: 4 second(s)

Memory limit: 262144 kilobytes

input: standard

output: standard

The notion of bidirected graphs is used to formulate and solve wide range of combinatorial

optimization problems: matchings, b-matchings, T-joins, T-paths packing and so on. Bidirected

graph is a generalization of directed graph, where every arc has two orientations — one for every

end.

More formally, bidirected graph is a tuple (V, E, ends, ω). V is a set of nodes, E is a set of edges,

ends is a mapping from E: ends(e) = {u, v}, where u, v ∈ V, . For every e ∈ E and v ∈

ends(e) we define ω(e, v) ∈ {-1, +1}. It is called the direction of the edge e in the node v.

An s-t walk in a bidirected graph G is an alternating sequence

P = (s = v0, e1, v1,..., ek, vk = t),

where vi ∈ V, ei ∈ E, ends(ei) = {vi - 1, vi} and ω(ei + 1, vi) ω(ei, vi) = -1.

A directed graph without loops can be considered as bidirected, if we replace each arc a = (u, v)

with an edge ea, and put ends(ea) = {u, v}, ω(ea, u) = -1, ω(ea, v) = +1.

An elementary transformation on node v is defined as follows: if for e ∈ E, ω(e, v) is defined,

then negate it. It's clear that elementary transformations don't change the set of walks in G.

We want to determine if it is possible to convert bidirected graph to "directed" graph (i.e. to

graph that can be obtained from some directed graph by means of the consideration described

above) using several elementary transformations. If it is, we also want to know how to do it

using the minimal number of elementary transformations.

Input The first line of the input contains two integer numbers n and m — the number of vertices and

the number of edges in the graph ( , ). Next m lines describe

the edges of the graph. Each line consists of four numbers ai, bi, di,1, di,2, where ai and bi are the

ends of the egde e and di,1=ω(e,ai), di,2=ω(e,bi); 1 ≤ ai, bi ≤ n, di,j ∈ {-1, 1}.

Output If it is impossible to perform required transformation, output should contain exactly one string "

NO

" (quotes for clarity only). Otherwise, the first line should contain string "

YES

", the second line should contain the number of elementary transformations in your solution, and

the following lines should describe those transformations. Each transformation should be

described by exactly one number — the index of the vertex of this transformation. If there are

several answers with the minimal number of transformations, output any of them.

Example(s)

sample input sample output

4 3

1 3 -1 -1

2 3 -1 -1

3 4 1 1

YES

1

3

sample input sample output

3 3

1 2 1 1

2 3 1 1

1 3 1 1

NO